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Mirrors > Home > HSE Home > Th. List > sh0 | Structured version Visualization version GIF version |
Description: The zero vector belongs to any subspace of a Hilbert space. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sh0 | ⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issh 28987 | . . 3 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻))) | |
2 | 1 | simplbi 500 | . 2 ⊢ (𝐻 ∈ Sℋ → (𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻)) |
3 | 2 | simprd 498 | 1 ⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ⊆ wss 3938 × cxp 5555 “ cima 5560 ℂcc 10537 ℋchba 28698 +ℎ cva 28699 ·ℎ csm 28700 0ℎc0v 28703 Sℋ csh 28707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-hilex 28778 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-sh 28986 |
This theorem is referenced by: ch0 29007 hhssabloilem 29040 hhssnv 29043 oc0 29069 ocin 29075 shscli 29096 shsel1 29100 shintcli 29108 shunssi 29147 omlsii 29182 sh0le 29219 imaelshi 29837 shatomistici 30140 |
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